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Pattern-equivariant homology

James J Walton

Algebraic & Geometric Topology 17 (2017) 1323–1373

Pattern-equivariant (PE) cohomology is a well-established tool with which to interpret the Čech cohomology groups of a tiling space in a highly geometric way. We consider homology groups of PE infinite chains and establish Poincaré duality between the PE cohomology and PE homology. The Penrose kite and dart tilings are taken as our central running example; we show how through this formalism one may give highly approachable geometric descriptions of the generators of the Čech cohomology of their tiling space. These invariants are also considered in the context of rotational symmetry. Poincaré duality fails over integer coefficients for the “ePE homology groups” based upon chains which are PE with respect to orientation-preserving Euclidean motions between patches. As a result we construct a new invariant, which is of relevance to the cohomology of rotational tiling spaces. We present an efficient method of computation of the PE and ePE (co)homology groups for hierarchical tilings.

aperiodic order, tilings, quasicrystals, tiling cohomology
Mathematical Subject Classification 2010
Primary: 52C23
Secondary: 37B50, 52C22, 55N05
Received: 10 February 2014
Revised: 22 September 2016
Accepted: 25 October 2016
Published: 17 July 2017
James J Walton
Department of Mathematics
University of York
YO10 5DD
United Kingdom